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Eldritch characters can be rather tricky to write, because writing them requires thinking outside of how a human might percieve things. Fiction where you're writing from a non-human's perspective is actually called "Xenofiction", from the Latin word "Xenos" meaning "foreign" and the word "fiction" meaning... fiction. XD Xenofiction can comprise anything from writing from the perspective of a dog, an elf, a dragon, or yes, even Eldritch entities.
To help clarify all this, I will use Slenderman, since I use him a lot in my writing and he kind of "speaks" to me the most of any of the Fears. I won't say I "understand" him best, because I'm constantly learning and developing new attributes to him as befitting a Fear of the Unknown, he's consistently undefined and as such always changing for me. But I do have some constants to which he tends to stick, and his themes resonate with what I tend to write most, if that makes sense.
The most important thing to remember is that eldritch beings, despite being very clearly not human or animal, still have personalities to a point. Sometimes they are not the same as a human personality might percieve them. For example, Slenderman stalks and sometimes kills humans. His motives for doing this might be as simple as "he's evil" or "for funsies", yes, but it could also reflect a desire to amuse himself, curiosity of humans and how they work, or even an urge to hunt and humans being the current prey du jour. Some adaptations have him eat humans somehow. Some don't. But that's a bit of a different issue and your portrayal would depend on your headcanon for Slenderman. Personally, I generally write Slenderman one of two ways: Either he is an extremely curious creature, ever running observations and experimental 'tests' on humankind, or he's this being that honestly sees his actions as 'play' and doesn't consider the impact it has on humans, good or bad - their opinions are quite irrelevant and in fact unintelligible to him, much as his would be to us. In both cases he tends to be quite an introverted, introspective, aloof, and intelligent soul, equally capable of being ruthless and cold as he is of being gentle and merciful, should he choose it. However his standards of what "ruthless" and "gentle" are differ wildly from what most humans think they are, which makes him come across as this cold, evil, dark, and dangerous predator. Oftentimes, to the confusion of his human targets, he comes across as completely unreadable, making him quite unknowable and frightening to humans in general.
It's always best to decide a motive for a character. That motive may or may not reflect the moral positions of "good" or "evil". For example, Slenderman might be chasing your character because that character knows something they shouldn't. He might also be after your character to warn them of something, to acquire something they have (perhaps knowledge of something), to kill them because he dislikes your character or otherwise, because he assumes stalking is a sign of friendship/'liking someone', because he's interested in their reactions, or because he thinks they'd make a good Proxy. All of these could be construed as "good" or "evil" if you really want, but they don't have to be. Many eldritch beings actually exhibit what we call "Blue and Orange Morality", that is, their morality does not reside on the human "good vs. evil" spectrum, even if humans percieve it as that. This can be tough to define, but I find the best approach is to simply portray the being in question as True Neutral - that is, they have no allegiance to anyone other than themselves and don't really bother with morality as we conceive of it. I often write Slenderman in this way, and in fact it's how I view all of the Fears. You can translate this to Fears being like a "force of nature" or something that doesn't have an alignment. The emotion of fear itself is a good example - it can be a good or bad thing, depending on circumstance. Regardless of the case, try to decide on a motive for your character first, and morality second - and then build the character's actions around the motive. And if you do use "good and evil" as a thing for eldritch morality, consider that since not all humans are fully good or evil by definition, neither are all eldritch beings.
Another thing to remember is that characters act and react to things. Slenderman might decide to kill the protagonist, but make a Proxy out of their friend. You of course don't have to reveal why this is in the prose, so long as "why" is set in your head somewhere. Maybe Slendy made a Proxy out of the friend because they are more physically or mentally adept, or because they are more receptive to his commands somehow. Maybe the protagonist is killed as a sort of "culling of the herd" of humanity, or is a necessary evil to achieve some goal or keep some event set in stone. Maybe they were killed because they simply made the wrong choice, or because it was a mercy killing. The possibilities are really pretty infinite, if you think outside of the box a bit. Another scenario: let's say a protagonist has discovered something that causes a snag in some plan of Slendy's. Now Slenderman, as a character that acts and reacts, must reconsider his course of action, and change how he reaches his goal based upon that. Characters act on their surroundings and on other characters, and react to events in their surroundings as well as the actions of other characters. Remember that, and try to keep your eldritch characters dynamic so as to keep plot moving forward.
Eldritch beings often experience the world in ways humans do not. Slenderman lacks all the facial sensory organs we humans have - he can't and doesn't necessarily smell, see, hear, or feel things the way we can. This is where your imagination gets to come into play. Perhaps his tentacles are very sensitive to vibrations or heat signatures given off by living beings, and that's how he hunts. Perhaps he echolocates and "sees" that way. Maybe he sees in ultraviolet (which damages normal human eyes over time, which would mean he has to see with a different method other than an eye). If he needs to eat, how does he do so? Via a concealable, alien 'mouth' of sorts? Does he use his tentacles like roots to suck nutrients from victims, or does he extend them and use them like plants use leaves, to photosynthesize? Does he feed upon human fear instead? Does he "see" sounds or "feel" color (synesthesia)? All this stuff can determine how Slenderman acts and reacts to an environment, another character, and events in the story. This is all dependent upon how you feel something without any eyes, nose, mouth, or ears to speak of might percieve the world and hunt prey, and it's based upon the character's actions and reactions in other portrayals of him.
It's the same for any other Fear - how has Plague Doctor been written in the past? If he's a mass of sentient bacteria, how do you think such a creature might percieve the world? Does Wooden Girl use her strings to pick up vibrations and catch people that way, maybe using them like a spider web to snag people? Is Cold Boy made of sentient ice, or does he cause ice to occur where he appears? Look at how others have portrayed these characters, and think about why they may have been portrayed that way. Character studies are always a great way to practice this - and that's actually exactly what "I, Slenderman" was, a way to get into his suit and tie and figure out what went on in that alien head of his. It was also a way to break myself of some of the fear I initially felt getting into that Mythos. Practice a bit with your characters and characters you like to get a feel for them. Who knows? Maybe you'll find a few Fears that "speak" to you and play well with you the way Slendy does with me. And of course, this being a highly open-source Mythos, all of this can change even between your own stories. You never have to write a Fear the same way twice! For example, in Twelve Days Slenderman is a mostly helpful, if highly frightening, living representation of anxiety and worry, whereas in areas of Project Him's stories, he's much more of a bogeyman-like figure, a monster in the closet, something that seeks to cause and spread as much fear as possible. And in Observation, he returns more to his original roots as an organ-snatching, forest-dwelling monster that kills for the sake of watching humans who know about him suffer and die. Three very different interpretations, yet all the same character.